Metamath Proof Explorer


Theorem wfii

Description: The Principle of Well-Ordered Induction. Theorem 6.27 of TakeutiZaring p. 32. This principle states that if B is a subclass of a well-ordered class A with the property that every element of B whose inital segment is included in A is itself equal to A . (Contributed by Scott Fenton, 29-Jan-2011) (Revised by Mario Carneiro, 26-Jun-2015)

Ref Expression
Hypotheses wfi.1 𝑅 We 𝐴
wfi.2 𝑅 Se 𝐴
Assertion wfii ( ( 𝐵𝐴 ∧ ∀ 𝑦𝐴 ( Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵𝑦𝐵 ) ) → 𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 wfi.1 𝑅 We 𝐴
2 wfi.2 𝑅 Se 𝐴
3 wfi ( ( ( 𝑅 We 𝐴𝑅 Se 𝐴 ) ∧ ( 𝐵𝐴 ∧ ∀ 𝑦𝐴 ( Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵𝑦𝐵 ) ) ) → 𝐴 = 𝐵 )
4 1 2 3 mpanl12 ( ( 𝐵𝐴 ∧ ∀ 𝑦𝐴 ( Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵𝑦𝐵 ) ) → 𝐴 = 𝐵 )